There are only 23=823=8 possible ways to arrange the genders boy/girl, with repetition. Since the probability of boy and girl are equal, the probability of each of these arrangements are also equal. We can therefore count the number of possible "good" answers, and divide by 8.
There are 3 possible ways to have 1 boy, that being "first/second/third child is boy", so the probability of exactly 1 boy is indeed 3838.
Having at most 2 girls is the opposite of having three girls, so since there are only one way of having three girls, there must be 8−1=78−1=7 ways of having at most two girls, so the probability of at most two girls is indeed 78
